(Apologies to Quantum Field Theorist for the gross simplification below.)
I'm going to assume you know about basic calculus and Taylor series; if not, I'll explain what these are in a subsequent answer if you are interested.
In Quantum Field Theory, we cannot do exact calculations ... only (better and better) approximations. The way we do these calculations is very much similar to doing a Taylor series in Calculus. However, the calculations involved many complicated multi-dimensional integrals
Feynman found out a way to represent these complicated mathematical expressions that occured in those approximate calculations as pictures (known as Feynman diagrams). In these pictures, one could represent (real) particles interacting as lines (straight, curvy, etc., depending on the rules) that end in open space..
As we increased the order of expansion in the approximation (think of powers of the variable in a Taylor series), the terms corresponded to more and more diagrams, each increasing in complexity with the order of expansion.
Well, it turned out that, using the Feynman rules for translating mathematical expressions into parts of a pictures, that lines (straight, curvy) similar to that of "real" particles appeared inside the diagrams; however, these lines were all connected to others and did not end in open space: they do not correspond to any particles that we can measure directly.
However, we can think of these lines as representing "virtual" particles, that we cannot probe directly.
One thing to keep in mind (and that many that mention the "virtual particle" explanations often forget to mention): these diagrams are, in a very strong sense, an artifact of the way we do calculations using the best approximation techniques we know. These are useful computational tools. But individual diagrams should not be thought of as representing an actual precise interaction that is taking place. And virtual particles are just a way to make a connection between what we think we know ("real" particles interacting) and what our approximative way of calculating things give us.
